I'm still pretty novice at Pet Battling, and after reaching 10 (and then 11 on a few members of my current go-to team), I am noticing that I don't seem to have in-battle access to their level 10 abilities.

I am assuming this will require swapping the level 10 ability with their level 1 ability?

If so, how do I do this? The UI is so horribly designed and unintuitive that no matter what I do or try, I can't figure it out.

Go to the Pet JournalMake sure the pet whose skills you wish to swap is in 1 of the 3 battleslotsThen hover over the pet's abilities in the battleslot window (in your case the first skill)Click this skill, and a dropdown menu will appear, allowing you to choose skill 1 or skill 2 (level 1 or level 10 skill)Click the new skill, and there you go, you have the other skill ready to use.[Hint: While you can switch skills in combat, the skill used in that battle won't change. The skill you go into battle with is the skill you're stuck with until the end of battle]

maybe your UI is bugged cause i find it well designed and intuitive xD

if that's the case ask a GM. if its not: click on the pet, then click on his lvl 1 ability and you will be given the option to choose between the lvl 1 and lvl 10 abilities (same for 2-15 and 4-20 abilities )